Rings with finite decomposition of identity
A criterion for semiprime rings with finite decomposition of identity to be prime is given. We also give a short survey on some finiteness conditions related to the decomposition of identity. We consider the notion of a net of a ring and show that the lattice of all two-sided ideals of a right semid...
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Dokuchaev, M.A. Gubareni, N.M. Kirichenko, V.V. 2020-02-18T04:19:17Z 2020-02-18T04:19:17Z 2011 Rings with finite decomposition of identity / M.A. Dokuchaev, N.M. Gubareni, V.V. Kirichenko // Український математичний журнал. — 2011. — Т. 63, № 3. — С. 319–340. — Бібліогр.: 48 назв. — англ. 1027-3190 https://nasplib.isofts.kiev.ua/handle/123456789/166010 512.5 A criterion for semiprime rings with finite decomposition of identity to be prime is given. We also give a short survey on some finiteness conditions related to the decomposition of identity. We consider the notion of a net of a ring and show that the lattice of all two-sided ideals of a right semidistributive semiperfect ring is distributive. An application of decompositions of identity to groups of units is given. Наведено критерiй первинностi напiвпервинних кiлець iз скiнченним розкладом одиницi, а також короткий огляд деяких умов скiнченностi вiдносно розкладу одиницi. Розглянуто поняття сiтки кiльця i показано, що решiтка всiх двобiчних iдеалiв правого напiвдистрибутивного напiвдосконалого кiльця є дистрибутивною. Наведено застосування розкладу одиницi до груп одиниць. en Інститут математики НАН України Український математичний журнал Статті Rings with finite decomposition of identity Кiльця iз скiнченним розкладом одиницi Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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| title |
Rings with finite decomposition of identity |
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Rings with finite decomposition of identity Dokuchaev, M.A. Gubareni, N.M. Kirichenko, V.V. Статті |
| title_short |
Rings with finite decomposition of identity |
| title_full |
Rings with finite decomposition of identity |
| title_fullStr |
Rings with finite decomposition of identity |
| title_full_unstemmed |
Rings with finite decomposition of identity |
| title_sort |
rings with finite decomposition of identity |
| author |
Dokuchaev, M.A. Gubareni, N.M. Kirichenko, V.V. |
| author_facet |
Dokuchaev, M.A. Gubareni, N.M. Kirichenko, V.V. |
| topic |
Статті |
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Статті |
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2011 |
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| container_title |
Український математичний журнал |
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Інститут математики НАН України |
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Article |
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Кiльця iз скiнченним розкладом одиницi |
| description |
A criterion for semiprime rings with finite decomposition of identity to be prime is given. We also give a short survey on some finiteness conditions related to the decomposition of identity. We consider the notion of a net of a ring and show that the lattice of all two-sided ideals of a right semidistributive semiperfect ring is distributive. An application of decompositions of identity to groups of units is given.
Наведено критерiй первинностi напiвпервинних кiлець iз скiнченним розкладом одиницi, а також короткий огляд деяких умов скiнченностi вiдносно розкладу одиницi. Розглянуто поняття сiтки кiльця i показано, що решiтка всiх двобiчних iдеалiв правого напiвдистрибутивного напiвдосконалого кiльця є дистрибутивною. Наведено застосування розкладу одиницi до груп одиниць.
|
| issn |
1027-3190 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/166010 |
| citation_txt |
Rings with finite decomposition of identity / M.A. Dokuchaev, N.M. Gubareni, V.V. Kirichenko // Український математичний журнал. — 2011. — Т. 63, № 3. — С. 319–340. — Бібліогр.: 48 назв. — англ. |
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2025-11-24T11:38:45Z |
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| fulltext |
UDC 512.5
M. A. Dokuchaev (Univ. São Paulo, Brazil),
N. M. Gubareni (Techn. Univ. Czȩstochowa, Poland),
V. V. Kirichenko (Kiev Nat. Taras Shevchenko Univ., Ukraine)
RINGS WITH FINITE DECOMPOSITION OF IDENTITY
КIЛЬЦЯ IЗ СКIНЧЕННИМ РОЗКЛАДОМ ОДИНИЦI
A criterion for semiprime rings with finite decomposition of identity to be prime is given. We also give a
short survey on some finiteness conditions related to the decomposition of identity. We consider the notion of
a net of a ring and show that the lattice of all two-sided ideals of a right semidistributive semiperfect ring is
distributive. An application of decompositions of identity to groups of units is given.
Наведено критерiй первинностi напiвпервинних кiлець iз скiнченним розкладом одиницi, а також ко-
роткий огляд деяких умов скiнченностi вiдносно розкладу одиницi. Розглянуто поняття сiтки кiльця i
показано, що решiтка всiх двобiчних iдеалiв правого напiвдистрибутивного напiвдосконалого кiльця є
дистрибутивною. Наведено застосування розкладу одиницi до груп одиниць.
1. Introduction. Recall that a ring is called semiprime if it does not contain nilpotent
nonzero ideals. A ring A is prime if a product of any two non-zero two-sided ideals of
A is not equal to zero.
A ring is said to be FDI-ring if it has a decomposition of identity into a finite sum of
pairwise orthogonal primitive idempotents. This class of rings includes right Artinian,
right Noetherian, semiperfect, and Goldie rings.
The article is organized as follows. In Section 2 we give a survey of some results and
examples on various finiteness conditions related to FDI-rings. In Section 3 a criterion
for semiprime FDI-finite rings to decompose into a finite direct product of prime rings
is given. From this main criterion we obtain corollaries which show that some classes
of semiprime rings decompose into a finite direct products of prime rings. Examples
of such classes of rings are right Bézout rings, right semidistributive rings, serial rings,
primely triangular rings, piecewise domains, right hereditary (semihereditary) FDI-rings
and right Noetherian right hereditary (semihereditary) rings. Section 4 is concerned with
generalized matrix rings. First the formula for the Jacobson radical for such rings is
given. Next, using the criterion for a decomposition of a semiprime ring obtained in
the previous section, we give some corollaries for a decomposition of some classes of
primely triangular matrix rings into a finite product of prime rings.
In Section 5 an application of decompositions of identity to Sylow subgroups in the
unit group is given.
In Section 6 we consider the notion of a net of a ring and use it to observe that the
lattice of all two-sided ideals of a semiperfect right semidistributive ring is distributive.
All rings in this paper are assumed to be associative (but not necessary commutative)
with 1 6= 0 and all modules are assumed to be unitary. The Jacobson radical of A will
be denoted by radA.
2. FDI-rings and finiteness conditions. In this section we give a survey of diverse
finiteness conditions related to that of FDI mentioned in Introduction.
An important role in the theory of rings and modules is played by various finiteness
conditions. In particular, minimal and maximal conditions, which can be formulated as
chain conditions on submodules and one-sided ideals. The minimal condition on right
(resp. left) ideals defines right Artinian (resp. left Artinian) rings. This condition is often
c© M. A. DOKUCHAEV, N. M. GUBARENI, V. V. KIRICHENKO, 2011
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 3 319
320 M. A. DOKUCHAEV, N. M. GUBARENI, V. V. KIRICHENKO
written as d.c.c. (descending chain condition) on right ideals. Analogously, right (resp.
left) Noetherian rings are defined as rings which satisfy the maximal condition, or the
a.c.c. (ascending chain condition), on right (resp. left) ideals.
In the formulations of the finiteness conditions in ring theory some special classes of
ideals are often chosen. One of the important classes of ideals is formed by the principal
right (left) ideals. The famous Theorem P by H. Bass [3] (see also [24], Theorem 10.5.5)
states that the d.c.c. on principal left ideals defines the right perfect rings.
Another important special class of ideals is formed by right (left) annihilators. One
can show that the set of all right (resp. left) annihilators of a ring A forms a lattice.
There is a lattice anti-isomorphism between the lattice of right annihilators of A and that
of left annihilators of A. This easily implies the next fact.
Proposition 2.1. The following conditions are equivalent for a ring A :
(1) A satisfies d.c.c. on right (resp. left) annihilators.
(2) A satisfies a.c.c. on left (resp. right) annihilators.
A. W. Goldie considered another finiteness condition. A module M is said to satisfy
the maximal condition for direct sums if M does not contain an infinite direct sum of
submodules [21]. A. W. Goldie proved that a module M with maximum condition for
direct sums has a finite uniform dimension u.dimM < ∞ (also called finite Goldie
dimension, or, simply, finite dimension)1. He also proved in [21] that a module M is
finite-dimensional if and only if M satisfies a.c.c. or d.c.c. on complements in M (recall
that a submodule C is called a complement in M if there exists a submodule S ⊆ M
such that C is maximal with respect to the property that C ∩ S = 0).
Summarizing the results obtained by Goldie in [21] we can write the following
theorem which shows the equivalence of different kind of finiteness conditions for a
module M.
Theorem 2.1. The following conditions are equivalent for a module M :
1. M is finite-dimensional.
2. u.dimM < ∞.
3. M satisfies a.c.c. on complements.
4. M satisfies d.c.c. on complements.
5. M contains no infinite direct sum of submodules.
One of the most frequently used finiteness condition for a ring A is connected with
the cardinal number of a set of pairwise orthogonal nonzero idempotents. Following to
A. A. Tuganbaev [42] we give the following definition.
Definition 2.1. A ring A is called orthogonally finite if it contains no infinite set
of pairwise orthogonal nonzero idempotents.
Using this definition we can write the following theorem:
Theorem 2.2 ([30], Proposition 6.59). For any ring A the following statements are
equivalent:
1. A is orthogonally finite.
2. A satisfies the ascending chain condition on right direct summands.
3. A satisfies the descending chain condition on left direct summands.
From the above theorem and Definition 2.1 we obtain the following examples of
orthogonally finite rings:
1. Semisimple rings.
1 The term “finite-dimensional” for such M was used originally by A. W. Goldie.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 3
RINGS WITH FINITE DECOMPOSITION OF IDENTITY 321
2. Noetherian rings.
3. Artinian rings.
4. Local rings (they have only one nonzero idempotent).
5. Rings with finite Goldie dimension.
6. Semiperfect rings. This follows by Theorem 10.3.7 from [24] which states that
a semiperfect ring is decomposed into a direct sum of right ideals each of which has
exactly one maximal submodule.
7. Perfect rings. This follows from Theorem P of H. Bass [3].
We proceed with the next:
Definition 2.2. A ring A is called Dedekind-finite (or von Neumann-finite, or
directly finite) if ba = 1 whenever ab = 1 for a, b ∈ A. Otherwise A is said to be
non-Dedekind-finite (or Dedekind-infinite). In this case there are elements a and b
such that ab = 1, but ba 6= 1.
The following easy fact gives simple examples of Dedekind-finite rings.
Proposition 2.2. A ring A is Dedekind-finite if it satisfies any of the following
conditions:
1. A is a unital subring2 of a Dedekind-finite ring.
2. A is a direct product of Dedekind-finite rings.
3. A is an epimorphic image of a Dedekind-finite ring.
4. A has no right or left zero divisors.
From this proposition and Definition 2.2 we immediately obtain the following
examples of Dedekind-finite rings:
1. A commutative ring.
2. A domain.
3. A division ring.
4. A ring which is a finite dimensional vector space over a division ring.
5. The ring of endomorphisms of a finitely dimensional vector space over a division
ring.
6. The ring of matrices Mn(D) over a division ring D.
7. Any finite product of Dedekind-finite rings.
8. A semisimple ring.
Definition 2.3. A ring A is called unit-regular if for any a ∈ A there exists a unit
u ∈ U(A) such that a = aua.
We shall use the following easy fact:
Lemma 2.1. Any unit-regular ring is Dedekind-finite.
Proof. Suppose ab = 1. Since there exists u ∈ U(A) such that a = aua, we obtain
1 = ab = auab = au which means that a = u−1 ∈ U(A), and so b = u ∈ U(A).
Therefore ba = 1, i.e., A is Dedekind-finite.
The following example shows the existence of non-Dedekind finite rings.
Example 2.1 (see [29, p. 4]). Let V be the k-vector space ke1 ⊕ ke2 ⊕ . . . with
a countably infinite basis {ei : i ≥ 1} over a field k, and let A = Endk(V ) be the
k-algebra of all vector space endomorphisms of V. Let a, b ∈ A be defined by
b(ei) = ei+1 for all i ≥ 1
and
2Recall that a subring A1 of A is said to be a unital subring if the identity of A is also the identity of A1.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 3
322 M. A. DOKUCHAEV, N. M. GUBARENI, V. V. KIRICHENKO
a(e1) = 0, a(ei) = ei−1 for all i ≥ 2,
then ab = 1 6= ba, so a is right-invertible without being left-invertible. Therefore A
gives an example of a non-Dedekind-finite ring.
With respect to the above example notice the next fact due to N. Jacobson [25].
Proposition 2.3 (N. Jacobson [25]). Any non-Dedekind-finite ring A contains a
countable infinite set of pairwise orthogonal nonzero idempotents in A.
The proof can be also found in [20] (Proposition 5.5), [29] (Proposition 21.26).
Proof. Let A be a ring which is non-Dedekind-finite, i.e., there exist elements
a, b ∈ A such that ab = 1 but ba 6= 1. Denote e = ba. Then e2 = b(ab)a = ba = e, so
e is a non-trivial idempotent. For i, j ≥ 0 let
eij = bi(1− e)aj .
Then {eij} is a set of matrix units in the sense that eijekl = δjkeil. To see this note that
aibi = 1 for all i, and a(1 − e) = 0 = (1 − e)b. If j 6= k, then ajbk is either a|j−k| or
b|j−k|, so
eijekl = bi(1− e)ajbk(1 − e)al = 0.
On the other hand, since 1− e is an idempotent,
eijejl = bi(1− e)ajbj(1− e)al = bi(1− e)al = eil.
Note that each eij 6= 0. Indeed, if bi(1− e)aj = 0, then 0 = aibi(1− e)ajbj = (1− e),
a contradiction.
In particular, {eii : i ≥ 1} is a countable infinite set of pairwise orthogonal nonzero
idempotents in A, and so A contains an infinite direct sum of nonzero right ideals
⊕
i≥0
eiiA.
Remark 2.1. From the proof of the above proposition it follows, in particular, that
there is a countable infinite set of pairwise orthogonal nonzero idempotents {eii : i ≥ 1}
in A such that A contains an infinite direct sum of nonzero right ideals
⊕
i≥0
eiiA.
Corollary 2.1. Let A be a ring with a.c.c. or d.c.c. on principal left (right) ideals
generated by idempotents. Then A is Dedekind-finite.
Corollary 2.2 (T. Y. Lam [29], Corollary 21.27). Let A be a ring such that B =
= A/radA is an orthogonally finite ring (i.e., does not contain an infinite direct sum of
nonzero right ideals). Then A is Dedekind-finite.
From this corollary and Proposition 2.2 we immediately obtain the following fact.
Corollary 2.3. A ring A is Dedekind-finite if and only if B = A/radA is Dedekind-
finite.
The next statement can be easily obtained from the above corollaries and Propositi-
on 2.3 and gives further examples of Dedekind-finite rings.
Proposition 2.4. A ring A is Dedekind-finite if it satisfies any of the following
conditions:
1. A has finite Goldie dimension.
2. A is right Noetherian.
3. A satisfies a.c.c. on right (left) direct summands.
4. A satisfies a.c.c. on right (left) annihilators.
5. A is a local ring.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 3
RINGS WITH FINITE DECOMPOSITION OF IDENTITY 323
6. A is a semilocal ring.
7. A is a semiperfect ring.
8. A is a perfect ring.
9. A is an orthogonally finite ring.
Proposition 2.5. (i) Let N be a nilpotent ideal of a ring A. Then A is Dedekind-
finite if and only if B = A/N is Dedekind-finite.
(ii) Let A be a Dedekind-finite ring. Then for any idempotent e ∈ A the ring eAe is
also Dedekind-finite.
(iii) Let A, B be rings and X be an A-B-bimodule. Then the ring
M =
(
A X
0 B
)
is Dedekind-finite if and only if A and B are both Dedekind-finite.
Proof. (i) The “only if” part follows from Proposition 2.2(3).
Suppose that A/N is Dedekind-finite and ϕ : A → A/N is the natural epimorphism.
Suppose ab = 1 for a, b ∈ A. Since A/N is Dedekind-finite, ϕ(ba) = ϕ(b)ϕ(a) =
= ϕ(a)ϕ(b) = ϕ(ab) = 1̄. Therefore ba = 1 + x, where x ∈ I. So it is invertible in A,
i.e., there exists u ∈ U(A) such that bau = 1, whence a = abau = au. Thus ba = 1, as
required.
(ii) Suppose that ab = e for a, b ∈ eAe. Put f = 1 − e. Since e, f are orthogonal
idempotents, (a+f)(b+f) = ab+f = e+f = 1. Taking into account that a+f, b+f ∈
∈ A, we obtain that (b+ f)(a+ f) = 1, whence ba = 1− f = e, as desired.
(iii) The proof follows immediately from (i) and (ii).
Consider the next important class of rings with a finiteness condition.
Definition 2.4 ([24], Chapter 2). A ring A is called an FDI-ring if there exists a
decomposition of the identity 1 ∈ A into a finite sum 1 = e1 + e2 + . . .+ en of pairwise
orthogonal primitive idempotents ei. In this case the regular A-module AA can be
decomposed into a finite direct sum of indecomposable modules eiA.
Note that the decomposition of 1 ∈ A, given in the definition of an FDI-ring, may
not be unique.
The following are important examples of FDI-rings:
1. Division rings.
2. Finite direct sums of FDI-rings.
3. Rings which are finite dimensional vector spaces over a division ring.
4. The rings of matrices Mn(D) over division rings D.
5. Semisimple rings.
6. Artinian rings.
7. Noetherian rings.
8. Semiperfect ring.
9. Rings with finite Goldie dimension.
10. Orthogonally finite rings.
11. Perfect rings.
The following statement is obvious.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 3
324 M. A. DOKUCHAEV, N. M. GUBARENI, V. V. KIRICHENKO
Proposition 2.6. Let A, B be rings and X be A-B-bimodule. If A and B are both
FDI-rings then the ring
M =
(
A X
0 B
)
is also FDI.
Proposition 2.7 ([24], Chapter 2). (i) Let A be an FDI-ring. Then the identity
of A can be written as a sum of a finite number of orthogonal centrally primitive
idempotents3.
(ii) Any FDI-ring can be uniquely decomposed into a direct product of a finite number
of indecomposable rings.
From the above we obtain the following diagram of a relationships between the
main classes of rings with finiteness conditions. Arrows in the diagram below mean
containments of classes of rings.
Dedekind-
finite
FDI
Orthogonally
finite
Noetherian semiperfect
Artinian
Next we show that there exist FDI-rings which are not Dedekind-finite.
Theorem 2.3 (J. C. Shepherdson [39]). There exists a ring R with 1 and no divisors
of zero over which there exist 2 × 2-matrices A, B, X such that AB = I, AX = 0,
X 6= 0, where I is the unit 2× 2-matrix.
It follows from this theorem that there exist a ring R with 1 and matrices A, B ∈
∈ M2(R) such that AB = I and BA 6= I. Indeed, otherwise if BA = I, then
BAX = 0 and IX = X = 0. Therefore the ring M2(R), where R as in Theorem 2.3,
3Recall that a central idempotent is called centrally primitive if it can not be decomposed into a sum of
orthogonal central non-zero idempotents.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 3
RINGS WITH FINITE DECOMPOSITION OF IDENTITY 325
is not Dedekind-finite. At the same time it is an FDI-ring, since 1 = e11 + e22, and,
moreover, e11M2(R)e11 ≃ e22M2(R)e22 ≃ R, where R is a domain.
Thus there are FDI-rings, which are non-Dedekind-finite. On the other hand the
following theorem holds.
Theorem 2.4 (T. Y. Lam [30], Proposition 6.60). Assume that a ring A does not
contain an infinite direct sum of nonzero right ideals. Then
(1) A is an FDI-ring.
(2) A is a Dedekind-finite ring.
Remark 2.2. The converse statement of this theorem is not true. The example
constructed by K. R. Goodearl in [20] (Example 5.15) shows that there exist unit-
regular rings (and therefore, by Lemma 2.1, Dedekind-finite rings) which contain un-
countable direct sums of nonzero pairwise isomorphic right ideals. Moreover, an example
constructed by L. A. Skornyakov in [36] shows that there exist FDI-rings which contain
an infinite set of pairwise orthogonal idempotents, and thus contain an infinite direct set
of principal right (left) ideals generated by idempotents.
For the proof of a generalization of the Krull – Remak – Schmidt – Azumaya theorem
P. Crawley and B. Jónsson [14] introduced the following definition.
Definition 2.5. Given a cardinal N, an A-module M is said to have the N-
exchange property if for any A-module X and any two decompositions of X :
X = M1 ⊕N =
⊕
i∈I
Xi
with M1 ≃ M and |I| ≤ N, there are submodules Yi ⊆ Xi such that
X = M1 ⊕
(
⊕
i∈I
Yi
)
.
A module M has the exchange property if M has the N-exchange property for any
cardinal N.
In [46] R. B. Warfield, Jr. introduced an exchange ring A as a ring whose left regular
module AA has the exchange property, and he also proved that this definition is left-right
symmetric. This class of rings includes (von Neumann) regular rings, local rings and
semiperfect rings. He also proved the following statement.
Proposition 2.8 (R. B. Warfield, Jr. [45]). An indecomposable module has the
exchange property if and only if its endomorphism ring is local.
This proposition immediately implies:
Proposition 2.9 (V. Camillo, H.-P. Yu [11]). Let A be an exchange ring. Then the
following statements are equivalent:
(1) A is an orthogonally finite ring;
(2) A is an FDI-ring;
(3) A is a semiperfect ring.
3. Semiprime FDI-rings. In this section we prove the main theorem which gives
a criterion for a semiprime FDI-ring to be decomposable into a finite direct product of
prime rings. This theorem can be considered as a generalization of the following theorem
proved by V. V. Kirichenko and M. Khibina [28] (see also [24], Theorem 14.4.3):
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 3
326 M. A. DOKUCHAEV, N. M. GUBARENI, V. V. KIRICHENKO
Theorem 3.1. A semiprime semiperfect ring is a finite direct product of indecompo-
sable rings. An indecomposable semiprime semiperfect ring is either a simple Artini-
an ring or an indecomposable semiprime semiperfect ring such that all its principal
endomorphism rings are non-Artinian.
The following proposition is well known (see [24], Proposition 11.2.4).
Proposition 3.1. Let A be a ring. The following conditions are equivalent:
1. A has no nonzero nilpotent ideal.
2. A has no nonzero nilpotent right ideal.
3. A has no nonzero nilpotent left ideal.
4. The prime radical of A is equal to zero.
A ring which satisfies any of the equivalent conditions of the above proposition is
called semiprime.
Lemma 3.1. Let J be an ideal of an FDI-ring A with a decomposition of the
identity 1 ∈ A into a finite sum 1 = e1 + e2 + . . .+ en of pairwise orthogonal primitive
idempotents ei. Then J is nilpotent if and only if ekJek is a nilpotent ideal in ekAek
for any k = 1, 2, . . . , n.
The proof of this lemma is exactly the same as that of Theorem 11.4.1 in [24]. It
also can be obtained by induction using the following lemma:
Lemma 3.2 (L. H. Rowen [33], Lemma 2.7.13). Let J be an ideal of a ring A.
Then J is nilpotent if and only if eJe and (1 − e)J(1 − e) are nilpotent for any
idempotent e ∈ A.
Lemma 3.3. Let A be a semiprime ring. Suppose 1 = g1 + g2 is a sum of
two idempotents, A =
(
A1 X
Y A2
)
, where A1 = g1Ag1, A2 = g2Ag2, X = g1Ag2
and Y = g2Ag1. Let M =
(
M11 M12
M21 M22
)
be an ideal in A and M12 6= 0. Then
M12M21 6= 0, M21 6= 0, M21M12 6= 0.
Proof. Suppose M12 6= 0. We shall show that M12M21 6= 0.
Assume M12M21 = 0. Denote by I(M12) the two-sided ideal in A generated by
M12. Since
(
0 M12
0 0
)(
A1 X
Y A2
)
=
(
M12Y M12
0 0
)
and
(
A1 X
Y A2
)(
M12Y M12
0 0
)
=
(
M12Y M12
YM12Y YM12
)
,
we obtain that I(M12) =
(
M12Y M12
YM12Y YM12
)
.
Obviously, M12YM12 ⊆ M12 and YM12Y ⊆ M21. Therefore
(M12Y )3 = (M12YM12)(YM12Y ) ⊆ M12M21 = 0.
Consequently, M12Y is nilpotent. In addition YM12YM12YM12 ⊆ YM12M21Y = 0.
Therefore YM12 is also nilpotent and, by Lemma 3.2, I(M12) is a nilpotent ideal.
The rest of the statement is verified analogously.
Lemma 3.3 is proved.
The following lemma is obvious.
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RINGS WITH FINITE DECOMPOSITION OF IDENTITY 327
Lemma 3.4. Let A be as in Lemma 3.3. Then
(
XY X
Y Y X
)
is a two-sided ideal
in a ring A.
Theorem 3.2. Let A be a semiprime FDI-ring, and suppose that 1 ∈ A have
the following decomposition into a sum of pairwise orthogonal primitive idempotents:
1 = e1 + . . . + en. A ring A is a finite direct product of prime rings if and only if all
rings eiAei are prime.
Proof. Let A be a semiprime ring, and let 1 = e1 + . . .+ en be a decomposition of
the identity of A into a sum of pairwise orthogonal primitive idempotents. Obviously it
suffices to prove only the converse statement.
Suppose that all eiAei are prime rings. We use induction on n.
1. If n = 1 the theorem is obvious.
2. Let n = 2. Suppose 1 = e1 + e2 and A =
(
A1 X
Y A2
)
. One can assume that
A is an indecomposable ring. Suppose X = 0. If Y 6= 0 then
(
0 0
Y 0
)
is a nonzero
nilpotent ideal. This is impossible, since A is semiprime. So Y = 0. But this is also
impossible, since A is indecomposable. Analogously one can prove that Y = 0 implies
X = 0. Thus if A is indecomposable then X 6= 0 and Y 6= 0.
Next we prove that A is prime.
Let L =
(
L1 L12
L21 L2
)
and M =
(
M1 M12
M21 M2
)
be two-sided ideals of A, and
LM = 0. Then L1M1 = 0 and L2M2 = 0. Since A1 and A2 are prime, it follows that
L1 = 0 or M1 = 0, and L2 = 0 or M2 = 0.
Assume that L 6= 0. If L1 = 0 and L2 = 0, then L is a nonzero nilpotent ideal by
Lemma 3.2. Therefore L1 6= 0 or L2 6= 0. Without loss of generality one can assume
that L1 = 0 and L2 6= 0. In this case M2 = 0 and M =
(
0 M12
M21 M2
)
is a two-sided
ideal in A. Hence M2 ⊆ M and thus we obtain that M12M21 = 0. Then, by Lemma 3.3,
M12 = 0 and M21 = 0. Consequently, M =
(
0 0
0 M2
)
. If M2 = 0, then M = 0 and
so A is prime.
Suppose M2 6= 0. Since M is a two-sided ideal in A, MA = M and from the
equality
(
0 0
0 M2
)(
A1 X
Y A2
)
=
(
0 0
M2Y M2
)
it follows that M2Y = 0. So, M2Y X = 0 in A2, which implies that Y X = 0, since A2
is prime and M2 6= 0. But the equality Y X = 0 contradicts to Lemma 3.3. Therefore,
M2 = 0 and if L 6= 0 then M = 0.
3. Let n ≥ 3. Denote g1 = e1 + . . .+ en−1, g2 = en. By the induction hypothesis
one can assume that the ring A1 = g1Ag1 is a finite direct product of prime rings, and
A2 = g2Ag2 is prime.
Case a. Assume that A1 = g1Ag1 is prime. If A is a decomposable ring, then the
statement is proved. If A =
(
A1 X
Y A2
)
is indecomposable, then one can apply a proof
similar to that in the previous case.
Case b. Let A1 = C1×. . .×Ct, where t ≥ 2 and each Ci is prime, for i = 1, . . . , t.
If A is indecomposable, then there exists a 3 × 3-minor eAe, where e = f1 + f2 + f3,
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328 M. A. DOKUCHAEV, N. M. GUBARENI, V. V. KIRICHENKO
f1, f2 ∈ {e1, . . . , en−1} and f3 = en,
B = eAe =
B1 0 B13
0 B2 B23
B31 B32 B3
,
with B1 = f1Af1, B2 = f2Af2, B3 = f3Af3, B13 = f1Af3, B23 = f2Af3, B31 =
= f3Af1, B32 = f3Af2 and all Bij are nonzero. Since each B1, B2 is one of the Ci,
i = 1, 2, . . . , t, and B3 = A2, all Bi are prime rings.
Let I(B13) be the ideal generated by B13 6= 0 and I(B23) the ideal generated by
B23 6= 0. Then
I(B13) =
B13B31 0 B13
0 0 0
B31B13B31 0 B31B13
and
I(B23) =
0 0 0
0 B23B32 B23
0 B32B23B32 B32B23
.
Obviously, I(B13)I(B23) = 0. Therefore, (B31B13)(B32B23) = 0. Since B3 is prime,
B31B13 6= 0 implies B32B23 = 0. But B23 6= 0, therefore it follows from Lemma 3.3
that B32B23 6= 0. This contradiction shows that either A1 is indecomposable, or A is
decomposable and A = C1 × . . .× Ct ×A2, where A2, Ci, i = 1, 2, . . . , t, are prime.
Remark 3.1. The proof of this theorem was inspired by Corollary 2.3.12 given in
[33], which states that a semiprime FDI-ring A is a semisimple Artinian. This statement
is not correct as shown by the obvious example of a discrete valuation ring O.
Proposition 3.2. A right uniserial semiprime ring O is prime.
Proof. Let L and N be nonzero two-sided ideals of O and LN = 0. One can
assume that N ⊆ L. Therefore N2 = 0, which implies that N = 0, since O is
semiprime. Consequently, O is a prime ring.
Definition 3.1. A ring A is said to be a right (resp. left) Bézout ring if every
finitely generated right (left) ideal of A is principal. A ring which is both right and left
Bézout is called a Bézout ring. A Bézout domain is a (commutative) integral domain in
which every finitely generated ideal is principal.
Any principal ideal ring is obviously Bézout. In a certain sense, a Bézout ring is a
non-Noetherian analogue of a principal ideal ring.
The main examples of commutative Bézout domains which are neither principal ideal
domains nor Noetherian rings are as follows.
1. The ring of all functions in a single complex variable holomorphic in a domain
of the complex plane C.
2. The ring of holomorphic functions given on the entire complex plane C.
3. The ring of all algebraic integers.
Corollary 3.1. A right Bézout semiprime semiperfect ring is a finite direct product
of prime rings.
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RINGS WITH FINITE DECOMPOSITION OF IDENTITY 329
Proof. The proof follows from Proposition 3.2 and [15] (Theorem 3.6).
Recall that a module M is called distributive if K ∩ (L +N) = K ∩ L +K ∩N
for all submodules K, L, N. A module is called semidistributive if it is a direct sum of
distributive modules. A ring A is called right (left) semidistributive if the right (left)
regular module AA (AA) is semidistributive. A right and left semidistributive ring is
called semidistributive.
The class of distributive (resp. semidistributive) modules contains the class of uni-
serial (resp. serial) modules.
We write an SPSDR-ring (SPSDL-ring) for a semiperfect right (left) semidistri-
butive ring and an SPSD-ring for a semiperfect semidistributive ring.
Theorem 3.3 ([19, 40, 41], Proposition 11.43). A semiperfect ring A is right (left)
semidistributive if and only if for any local idempotents e and f of A the set eAf is
a uniserial right fAf -module (uniserial left eAe-module). In particular, if e is a local
idempotent of a semiperfect right (left) semidistributive ring A then eAe is a uniserial
ring.
Corollary 3.2. Every semiprime SPSDR-ring is isomorphic to a finite direct pro-
duct of prime SPSDR-rings.
Proof. The proof follows from Proposition 3.2 and Theorem 3.3.
Since any right serial ring is an SPSDR-ring we obtain:
Corollary 3.3. Every right serial semiprime ring is isomorphic to a finite direct
product of prime serial rings.
Note that the last corollary is proved in [31] (Proposition 3.4).
4. Generalized matrix rings. Generalized matrix rings form one of the largest
classes of matrix rings and studied extensively in various contexts.
Let A, B be rings, M an A-B-bimodule, and N an B-A-bimodule. Assume that
there are two bimodule homomorphisms f : M ⊗B N → A and g : N ⊗A M → B,
satisfying the associativity conditions:
f(m⊗ n)m1 = mg(n⊗m1) and g(n⊗m)n1 = nf(m⊗ n1)
for all m,m1 ∈ M and n, n1 ∈ N.
Consider a set
Q =
(
A M
N B
)
of all matrices
(
a m
n b
)
, where a ∈ A, b ∈ B, m ∈ M and n ∈ N. Addition in this
set is defined componentwise and multiplication by the rule:
(
a m
n b
)(
a1 m1
n1 b1
)
=
(
aa1 + f(m⊗ n1) am1 +mb1
na1 + bn1 g(n⊗m1) + bb1
)
.
With respect to these operations Q becomes a ring, which is called a generalized matrix
ring (of order 2).
Note, that if we put m ·n = f(m⊗n) and n ·m = g(n⊗m) we obtain the operation
of multiplication as in the usual matrix ring.
In a similar way we can define a generalized matrix ring of any order n > 2. Such a
ring is naturally isomorphic to a generalized matrix ring of order 2 (this isomorphism is
obtained by partitioning every matrix ring into four blocks).
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330 M. A. DOKUCHAEV, N. M. GUBARENI, V. V. KIRICHENKO
Remark 4.1. If a ring A has a finite set of pairwise orthogonal idempotents
e1, e2, . . . , en such that 1 = e1 + e2 + . . .+ en, then its two-sided Peirce decomposition
in a usual way can be considered as a generalized matrix ring.
Lemma 4.1. Let A =
(
A1 X
Y A2
)
be a generalized matrix ring, where X is
an A1-A2-bimodule and Y is an A2-A1-bimodule. Suppose that XY ⊆ rad(A1) and
Y X ⊂ rad(A2). Then rad(A) =
(
R1 X
Y R2
)
, where R1 = rad(A1) and R2 =
= rad(A1).
Proof. Write 1 = e1 + e2, with eiAei = Ai, e1Ae2 = X, e2Ae1 = Y, and
R = rad(A). By [24] (Proposition 3.4.8.), Ri = eiRei for i = 1, 2. Write also
L =
(
R1 X
Y R2
)
.
Since, by assumption, XY ⊆ R1 and Y X ⊆ R2, L is a two-sided ideal in A.
We shall show that 1− u is invertible for any u ∈ L. Write
1− u =
(
1− r1 x
y 1− r2
)
.
Then
(
1− r1 x
y 1− r2
)(
1 −(1− r1)
−1x
0 1
)
=
(
1− r1 0
y 1− r′2
)
,
where r′2 = r2 + y(1− r1)
−1x ∈ R2. Furthermore,
(
1 0
−(1− r1)
−1y 1
)(
1− r1 0
y 1− r′2
)
=
(
1− r1 0
0 1− r′2
)
.
Since the latter matrix is clearly invertible, so too is 1− u.
By [24] (Proposition 3.4.6), L ⊆ radA, and by [24] (Proposition 3.4.8), radA ⊆ L.
Thus, radA = L, as required.
Lemma 4.1 is proved.
Theorem 4.1. Let
A =
A1 X12 . . . X1n
X21 A2 . . . X2n
...
...
. . .
...
Xn1 Xn2 . . . An
be a generalized matrix ring, where Xij is an Ai-Aj-bimodule i = 1, 2, . . . , n. Suppose
that XijXji ⊆ rad(Ai) for i = 1, 2, . . . , n, i 6= j. Then
rad(A) =
R1 X12 . . . X1n
X21 R2 . . . X2n
...
...
. . .
...
Xn1 Xn2 . . . Rn
,
where Ri = rad(Ai) for i = 1, 2, . . . , n.
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RINGS WITH FINITE DECOMPOSITION OF IDENTITY 331
Proof. The theorem follows by induction on n using Lemma 4.1.
Corollary 4.1. Let
A =
A1 X12 . . . X1n
0 A2 . . . X2n
...
...
. . .
...
0 0 . . . An
be a triangular matrix ring, where Xij is an Ai-Aj-bimodule, i = 1, 2, . . . , n. Then
rad(A) =
R1 X12 . . . X1n
0 R2 . . . X2n
...
...
. . .
...
0 0 . . . Rn
,
where Ri = rad(Ai) for i = 1, 2, . . . , n.
Definition 4.1. Let A be an FDI-ring. A decomposition of the identity 1 = f1 +
+ f2 + . . . + fm ∈ A, where fi are idempotents, is called triangular if fiAfj = 0
for all i > j. Such a decomposition is called prime if fiAfi is a prime ring for any
i = 1, . . . ,m.
Definition 4.2. An FDI-ring A is called primely triangular if there exists a
prime triangular decomposition of the identity of A. In this case the two-sided Peirce
decomposition of A can be represented in the following triangular form:
A =
A1 A12 . . . A1,n−1 A1n
0 A2 . . . A2,n−1 A2n
...
...
. . .
...
...
0 0 . . . An−1 An−1,n
0 0 . . . 0 An
, (1)
where all Ai are prime rings.
Remark 4.2. Note that the term “triangular ring” was first introduced by S. U. Chase
[12] in 1961 for semiprimary rings. According to S. U. Chase a semiprimary ring A with
Jacobson radical R is triangular if there exists a complete set e1, e2, . . . , ek of mutually
orthogonal primitive idempotents of A such that eiAej = 0 for all i > j. This notion
was used by L. W. Small in [38] for arbitrary right Noetherian rings. M. Harada [23] in
1966 introduced the term “generalized triangular matrix rings” for rings with triangular
decomposition of the identity where Ai = eiAei are arbitrary rings. He studied the
properties of such rings when all Ai are semiprimary rings. It is obvious that in this case
the generalized triangular matrix rings are also semiprimary. Yu. A. Drozd [17] in 1980
used the term “triangular ring” for rings with a triangular prime decomposition of the
identity.
The following simple fact is well known.
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332 M. A. DOKUCHAEV, N. M. GUBARENI, V. V. KIRICHENKO
Proposition 4.1 (see [24], Proposition 9.2.13). Let A be a prime (semiprime) ring
with an idempotent e2 = e ∈ A. Then eAe is also a prime (semiprime) ring.
Theorem 3.2 and the above proposition immediately give the next:
Corollary 4.2. Any semiprime primely triangular FDI-ring is isomorphic to a finite
direct product of prime rings.
Recall that a ring A is a piecewise domain (or simply PWD) with respect to a
complete set of pairwise orthogonal idempotents {e1, e2, . . . , en} if xy = 0 implies
x = 0 or y = 0 with x ∈ eiAej and y ∈ ejAek for 1 ≤ i, j, k ≤ n.
Note that [24] (Proposition 10.7.8) implies that the condition in the above definition
is equivalent to the following one: every nonzero homomorphism eiA → ejA is a
monomorphism for 1 ≤ i, j ≤ n.
Piecewise domains were first introduced and studied by R. Gordon and L. Small
in [22]. Given a piecewise domain A, each eiAei is a domain, i = 1, . . . ,m, and
hence the set of {e1, e2, . . . , em} is a complete set of primitive orthogonal idempotents.
Consequently, any piecewise domain is an FDI-ring. The structure of piecewise domains
was obtained by R. Gordon and L. Small in [22]. The next fact follows from their
description.
Proposition 4.2. A piecewise domain is a primely triangular ring.
Remark 4.3. Note that a piecewise domain can be considered as a generalization
of an l-hereditary ring. Recall that a ring A is called l-hereditary if it is Artinian and
satisfies the following condition: given any pair of indecomposable projective left A-
modules P and Q and any non-zero A-homomorphism ϕ : P → Q, either ϕ = 0 or ϕ
is a monomorphism. There is the following equivalent condition: any local submodule
of an indecomposable projective module is projective again (recall that a local module
is a module with a unique maximal submodule). Note that such algebras were first
considered by R. Bautista [4] and the term l-hereditary was introduced by R. Bautista
and D. Simson in [5] in their study of representations of incidence algebras of posets.
Assume that 1 = e1+ e2+ . . .+ en is a decomposition of the identity of a piecewise
domain A. If e is a sum of different idempotents from the set {e1, e2, . . . , en}, then the
ring eAe is again a piecewise domain. Therefore the above proposition and Theorem 3.2
give the next fact.
Corollary 4.3 (R. Gordon and L. Small [22]). Every semiprime piecewise domain
is isomorphic to a finite direct product of prime piecewise domains.
Since any right hereditary (semihereditary) FDI-ring A is a piecewise domain and
eAe is a hereditary (semihereditary) ring for any idempotent e ∈ A (see [34], Corollary 1)
we obtain the following:
Corollary 4.4. Every semiprime right hereditary (semihereditary) FDI-ring is iso-
morphic to a finite direct product of right hereditary (semihereditary) prime rings.
Since any right Noetherian right hereditary (semihereditary) ring A is a piecewise
domain and eAe is a right Noetherian right hereditary (semihereditary) ring for any
idempotent e ∈ A, we have the next:
Corollary 4.5. Every semiprime right Noetherian right hereditary (semihereditary)
ring is isomorphic to a finite direct product of right Noetherian right hereditary (semi-
hereditary) prime rings.
5. An application of the decomposition of the identity to the unit group. In this
section we give an improvement on a fact on Sylow subgroups of general linear groups
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RINGS WITH FINITE DECOMPOSITION OF IDENTITY 333
over finite local rings given in Corollary 7.6 in [16]. Elements of an FDI-ring A can be
represented by generalized matrices which makes it convenient to perform calculations
with them and, in particular, to obtain results on the group of units U(A) of A (see, for
example, [7 – 10]).
Our goal is to produce a Sylow p-subgroup in the unit group of a semiperfect ring
A assuming that the characteristic of A is a power of p and the Jacobson radical of A
is a nil ideal, and comment conjugacy. Recall that a semiperfect ring whose Jacobson
radical is nilpotent is called semiprimary. We shall say that a ring A is almost perfect
if A is semiperfect and its Jacobson radical is nil. Note that since any left (or right)
T -nilpotent ideal is nil, it follows that any left (or right) perfect ring is almost perfect
(see [3], Theorem P). In particular, examples from [3, p. 475, 476] show that there exist
almost perfect rings which are not semiprimary.
If D is a division ring and m > 0, we denote by UTm(D) the upper unitriangular
group, i.e., the group of those m×m-upper triangular matrices over D whose diagonal
entries are all equal to 1. Notice that UTm(D) contains all upper-triangular transvections,
i.e., the matrices of the form 1 + eij(x), where i < j; i, j ∈ {1, . . . ,m}, x ∈ D, and
eij(x) stands for the m×m-matrix whose unique non-zero entry is placed in the (i, j)-
position and is equal to x4. Multiplying any element g = (gij) of the general linear
group GLm(D) by 1+eij(x) from the left we add to the i-row of g the j-one multiplied
by any element x ∈ D from the left, and, similarly, multiplying g by 1 + eij(x) from
the right we add to the j-column of g the i-one multiplied by x from the right.
The next fact is known. It follows from [35, p. 17 – 19].
Lemma 5.1. Let D be a division ring of characteristic p > 0 which is locally
finite-dimensional over its center. Then any Sylow p-subgroup of the general linear group
GLm(D) is conjugate in GLm(D) to UTm(D).
The fact that UTm(D) is a Sylow p-subgroup of GLm(D) when D is any division
ring with characteristic p > 0, is rather elementary. For the sake of completeness we
give a proof below.
Lemma 5.2. Let D be a division ring whose characteristic is p > 0 and let m
be a positive integer. Then UTm(D) is a Sylow p-subgroup of the general linear group
GLm(D).
Proof. Suppose on the contrary that G̃ ⊆ GL(n,D) is a p-subgroup which contains
properly UTm(D), and take g = (gij) ∈ G̃ \ UTm(D). It is easily seen that g can be
chosen to have all entries in the first row equal to 0 except the (1, 1)-position. Indeed, if
g11 = 0 then some gk1 6= 0, as otherwise g can not be invertible. Adding to the first row
the k-one we come to the case g11 6= 0. Now for any i > 1, adding to the i-column the
first one multiplied by −g−1
11 · g1i from the right, we produce 0 in the (1, i)-entry. Since
i was arbitrary, we may suppose that g11 6= 0 and g1i = 0 for all i > 1, as desired.
Let G be the subgroup of G̃ formed by those matrices, all whose entries in the first
row, except the (1, 1)-one, are zero. Then the (1, 1)-entries of the elements from G form
a p-subgroup of units in D, and since D has no p-elements, we have that
h11 = 1 ∀h = (hij) ∈ G, (2)
in particular, g11 = 1. Furthermore, the lower-right (m − 1) × (m − 1) blocks of the
matrices of G form a p-subgroup in GL(m − 1,K) which contains UTm−1(D). By
4In fact, UTm(D) is generated by the upper triangular transvections.
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334 M. A. DOKUCHAEV, N. M. GUBARENI, V. V. KIRICHENKO
induction on m, it coincides with UTm−1(D). Then since g /∈ UTm(D), g must have a
non-zero non-diagonal entry in the first column, say g1l 6= 0, l > 1. Adding to the first
row of g the l-one multiplied by xg−1
1l from the left, we obtain an element g̃ ∈ G̃ with
arbitrary x ∈ D in the (1, 1)-position. Some non-diagonal entries of the first row of g̃ are
not 0 now. But they can be made equal to 0 as above using the non-zero (1, 1)-position.
Thus we come to an element h ∈ Ḡ whose (1, 1)-entry x is an arbitrary non-zero
element of D. If |D| > 2 this is a contradiction. The lemma for the case |D| = 2 is a
very well-known example from basic group theory.
Lemma 5.2 is proved.
Note that the structure and conjugacy of the Sylow subgroups of GLm(D), where
D is a division ring which is finite-dimensional over its center, were extensively studied
by A. E. Zalesski in [48].
Let A be a semiperfect ring with a fixed decomposition 1 = f1 + . . . + fs of the
identity into pairwise orthogonal idempotents such that
A = f1A+ . . .+ fsA = Pn1
1 ⊕ . . .⊕ Pns
s ,
where P1, . . . Ps are the pairwise non-isomorphic principal right A-modules. Note that
by [24] (Proposition 2.1.3) we have that each fiAfi is isomorphic to the full matrix ring
over the endomorphism ring of Pi :
fiAfi ∼= Mni
(EndA(Pi)), i = 1, . . . , s,
and, moreover, each EndA(Pi) = Oi is a local ring whose (unique) maximal ideal shall
be denoted by Mi. Then the two-sided Peirce decomposition of A has the following
form:
A =
Mn1
(O1) A12 . . . A1s
A21 Mn2
(O2) . . . A2s
...
...
. . .
...
As1 As2 . . . Mns
(Os)
, (3)
where Aij = fiAfj, i, j = 1, . . . , s. Since rad (Mn1
(O1)) = Mn1
(M1), we obtain by
Proposition 11.1.1 from [24] that the Jacobson radical R = radA of A has the following
two-sided Peirce decomposition:
R =
Mn1
(M1) A12 . . . A1s
A21 Mn2
(M2) . . . A2s
...
...
. . .
...
As1 As2 . . . Mns
(Ms)
. (4)
Notice that AijAji ⊆ Mns
(Ms), i = 1, . . . , s, i 6= j, as R is a two-sided ideal in A.
Let B be the subring of A formed by the radically upper-triangular matrices, i.e., the
subring whose two-sided Peirce decomposition has the following form:
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RINGS WITH FINITE DECOMPOSITION OF IDENTITY 335
B =
Hn1
(O1) A12 . . . A1s
A21 Hn2
(O2) . . . A2s
...
...
. . .
...
As1 As2 . . . Hns
(Os)
,
where
Hni
(Oi) =
Oi Oi . . . Oi
Mi Oi . . . Oi
...
...
. . .
...
Mi Mi . . . Oi
.
Since
AijAji ⊆ Mni
(Mi) ⊆
Mi Oi . . . Oi
Mi Mi . . . Oi
...
...
. . .
...
Mi Mi . . . Mi
= radHni
(Oi) ∀i 6= j,
we can apply Theorem 4.1, and consequently,
R′ = radB =
radHn1
(O1) A12 . . . A1s
A21 radHn2
(O2) . . . A2s
...
...
. . .
...
As1 As2 . . . radHns
(Os)
.
Then G = 1 + R′ is a normal subgroup of U(B). Note that if R′ is nilpotent, i.e.,
(R′)m = 0 for some m ≥ 0, then G = 1 +R′ is a nilpotent group and
1 +R′ ⊃ 1 + (R′)2 ⊃ . . . ⊃ 1 + (R′)m−1 ⊃ 1
is a central series of G.
Let A be an almost perfect ring. Notice that if the characteristic of each division ring
Di = Oi/Mi is the same prime p > 0 for all i = 1, . . . , s, then the characteristic of A
is a positive power of p. Evidently, the converse is also true. Keeping our notation we
give the next:
Theorem 5.1. Let A be an almost perfect ring whose characteristic is a power of
a prime p > 0. Then G is a Sylow p-subgroup of U(A). If, in addition, each division
ring Oi/Mi is locally finite-dimensional over its center, then all Sylow p-subgroups of
U(A) are conjugate to G.
Proof. It is easily seen that R′ nilpotent modulo R and, consequently, R′ is a nil
ideal as so too is R. Then the binomial formula implies that G is a p-group. Suppose
that G̃ is a p-subgroup of U(A) with G̃ ⊇ G. Let ϕ : A → A/R be the canonical map.
By (3) and (4) we have the following block-matrix decomposition:
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 3
336 M. A. DOKUCHAEV, N. M. GUBARENI, V. V. KIRICHENKO
A/R =
Mn1
(D1) 0 . . . 0
0 Mn2
(D2) . . . 0
...
...
. . .
...
0 0 . . . Mns
(Ds)
. (5)
Hence
U(A/R) ∼= GLn1
(D1)× . . .×GLns
(Ds) (6)
and
ϕ(G) ∼= UTn1
(D1)× . . .×UTns
(Ds).
It follows by Lemma 5.2 that ϕ(G̃) = ϕ(G), and, consequently, G̃ = G, as Kerϕ ⊆ G.
Assume now that each Di, i = 1, . . . , s, is locally finite-dimensional over its center.
Let ϕ be as above and
ϕ̄ : U(A) → GLn1
(D1)× . . .×GLns
(Ds)
be the corresponding homomorphism of the unit groups. Notice that Ker ϕ̄ = 1 +R is
a normal p-subgroup in U(A), as R is a nil ideal. It follows that any Sylow p-subgroup
G̃ of U(A) contains Ker ϕ̄. By Lemma 5.1 there exists a unit v ∈ ϕ̄(U(A)) such that
v−1ϕ̄(G̃)v = ϕ̄(G). Let u, u1 ∈ A be elements with ϕ̄(u) = v and ϕ̄(u1) = v−1. Then
uu1 = 1+ x ∈ 1+R, and since 1+R is a group, it follows that u1(1+ x)−1 is a right
inverse for u. Hence u is invertible, as A is Dedekind-finite (see Proposition 2.4). Finally,
thanks to the fact that both G̃ and G contain Ker ϕ̄, we conclude that u−1G̃u = G.
Theorem 5.1 is proved.
Corollary 5.1. Let A be a finite ring whose characteristic is a power of a prime
p > 0. Then any Sylow p-subgroup of U(A) is conjugate to G.
Observe that the conjugacy of the Sylow subgroups stated in the above corollary is
not reliant on Lemma 5.1 thanks to Sylow’s theorem.
Remark 5.1. Let A be as in Theorem 5.1 and ϕ̄ as above. Assume that q is a prime
different from p. Since Ker ϕ̄ is a p-group, it follows that H ∼= ϕ̄(H) for any q-subgroup
H in U(A).
6. Nets in rings. The notion of a net often appears in different fields of mathematics,
in particular, in algebra (see, for example, [1, 2, 7 – 9, 18, 43, 44]).
We shall consider the notion of a net in an associative ring A associated with a
decomposition of 1 into a sum of pairwise orthogonal idempotents.
Let
1 = e1 + . . .+ en (7)
be a fixed decomposition of 1 ∈ A into a sum of pairwise orthogonal idempotents,
and A = e1A ⊕ . . . ⊕ enA the corresponding decomposition of the ring A into a
direct sum of right ideals. For any element a ∈ A there results a = 1 · a · 1 =
= (e1 + . . . + en)a(e1 + . . . + en) =
∑n
i,j=1
eiaej. It is not difficult to verify that
such a decomposition defines a decomposition of the ring A into a direct sum of Abelian
groups eiAej , i, j = 1, 2, . . . , n:
A =
n
⊕
i,j=1
eiAej . (8)
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RINGS WITH FINITE DECOMPOSITION OF IDENTITY 337
The two-sided Peirce decomposition of A has the following form:
A =
A11 A12 . . . A1n
A21 A22 . . . A2n
...
...
. . .
...
An1 An2 . . . Ann
, (9)
where Aij = eiAej . Note that the elements of eiAej are naturally identified with
homomorphisms from ejA to eiA.
Definition 6.1. A direct sum
N =
n
⊕
i,j=1
Nij , (10)
where Nij is an Ai-Aj-subbimodule in Aij for all i, j = 1, . . . , n, is called a net in A
associated with (7).
The notion of a net associated with the trivial decomposition of 1 ∈ A coincides
with the notion of an ideal of A. Note if in the above definition we add the condition
that NijNjk ⊆ Nik, for all i, j, k, then we obtain the concept of a net subring introduced
in [10]
Let S be a poset and T a subset of S. An element a ∈ S is called an upper bound
(resp. lower bound) of T if t ≤ a (resp. a ≤ t) for all t ∈ T. In general a poset can
have several upper bounds or it can have none at all.
An element a ∈ T is a greatest (resp. least) element of T if t ≤ a (resp. a ≤ t)
for all t ∈ T. Not every subset T of a poset S has a greatest (or least) element. But if
T has such an element then it is unique. Indeed, let x and y be greatest elements of T.
Then x ≤ y and y ≤ x. Hence from the property of “antisymmetry” of a poset it follows
that x = y. The uniqueness of a least element of T can be proved analogously. So the
greatest (resp. least) element, if it does exist, is unique and is an upper (resp. lower)
bound for T. If the set of upper bounds of T has a least element, then it is called the
least upper bound ( or supremum) of T and denoted by sup(T ). If the set of lower
bounds has a greatest element, it is called the greatest lower bound (or infimum) of T
and denoted by inf(T ). It is obvious that if a subset T has a supremum (resp. infimum),
then it is uniquely determined.
Definition 6.2. A poset S, whose every pair of elements has both a supremum and
an infimum in S, is said to be a lattice.
Example 6.1. Let A be a ring and L(A) the set of all ideals of the ring A ordered
by inclusion. Let I and J be ideals in A. Then their supremum in L(A) is the sum
I +J and their infimum in L(A) is the intersection I ∩J . Therefore L(A) is a lattice.
Let S be a lattice. Then each pair of elements a, b ∈ S has both a supremum and an
infimum. Denote
a ∨ b = sup{a, b} and a ∧ b = inf{a, b}.
Then the maps ∨ and ∧ from S × S to S defined by
(a, b) 7→ a ∨ b and (a, b) 7→ a ∧ b
are binary operations on S.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 3
338 M. A. DOKUCHAEV, N. M. GUBARENI, V. V. KIRICHENKO
Let N(A) be a set of all nets in A associated with a fixed decomposition of the
identity of A. We can define on N(A) an ordering relation � by the following way: for
two sets N =
n
⊕
i,j=1
Nij and M =
n
⊕
i,j=1
Mij of N(A) we say that N � M if and only
if Nij ⊆ Mij for any pair i, j = 1, 2, . . . , n. On the set N(A) we can also define the
operations of addition and intersection in the natural way: N +M =
n
⊕
i,j=1
(Nij +Mij)
and N ∩M =
n
⊕
i,j=1
(Nij ∩Mij).
Proposition 6.1. The set N(A) of all nets associated with a fixed decomposition
of the identity of a ring A is a lattice with N1∨N2 = N1+N2 and N1∧N2 = N1∩N2,
where N1, N2 ∈ N(A).
The proof is obvious.
Definition 6.3. A lattice L is called distributive if :
x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ y) for all x, y, z ∈ L.
Consider the lattice L(A) of all two-sided ideals I of a ring A. This lattice is
distributive if I ∩ (K + L) = (I ∩K) + (I ∩ L) for all I, K, L ∈ L(A).
Lemma 6.1 [6, p.12]. Any chain is a distributive lattice and any direct product of
distributive lattices is a distributive lattice.
Let I be a two-sided ideal of A. Then for the decomposition (7) of the identity the
two-sided Peirce decomposition of I has the following form:
I =
I11 I12 . . . I1n
I21 I22 . . . I2n
...
...
. . .
...
In1 In2 . . . Inn
,
where each Iij ⊆ Aij is an Ai-Aj-module for i, j = 1, 2, . . . , n. Therefore any ideal I
is a net in A, and so L(A) ⊂ N(A).
Theorem 6.1. The lattice L(A) of a SPSDR-ring A is distributive.
Proof. Let A be a semiperfect right semidistributive ring, and let 1 = e1 + e2 + . . .
. . .+ en be a decomposition of the identity of A into a sum of pairwise orthogonal local
idempotents.
By [24] (Corollary 14.2.2), in this case any component Aij of the decomposition (9)
is a uniserial right Ai-module. Therefore any net N(i, j), which i, j-component is equal
to Nij ⊆ Aij and all others are equal to 0, is a chain. Hence from Lemma 6.1 it follows
that N(A), as a direct product of distributive lattices, is also a distributive lattice. The
lattice L(A) is distributive as a sublattice of a distributive lattice N(A).
Corollary 6.1. A lattice L(A) of a right Bézout semiperfect ring A is distributive.
Note that in the commutative case this corollary was proved for any Bézout ring
by Chr. U. Jensen [26] (indeed he proved this statement for any Prüfer ring) and by
P. M. Cohn [13].
Acknowledgements. The first author was partially supported by CNPq and FAPESP
of Brazil. The last two authors were supported by FAPESP of Brazil, and they thank the
Department of Mathematics of the University of São Paulo for its warm hospitality during
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 3
RINGS WITH FINITE DECOMPOSITION OF IDENTITY 339
their visit in 2010. The authors are grateful to Professors L. Small and E. Zelmanov for
useful conversations.
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Received 27.11.10
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